Secure a zone with robots
Luc Jaulin and Benoˆıt Zerr
Lab-STICC, ENSTA Bretagne, Brest, France lucjaulin@gmail.com
Problem. We consider n underwater robots R1, . . . ,Rn at po- sitions a1, . . . ,an and moving in a 2D world [1]. Each robot has a visibility zone. If an intruder is inside the visibility zone of one robot, it is detected. The robots have collaborate to guarantee that they is no moving intruder inside a subzone of a compact subset O of R2, representing the 2D ocean.
Complementary approach. We assume that there exists a vir- tual intruder moving inside O satisfying the differential inclusion
˙
x(t) ∈ F(x(t)),
where x(t) is the state vector. Moreover, we assume that each robot Ri has a visibility zone of the form g−1ai ([0, d]) where dis the scope. Our contribution is to show that characterizing the secure zone translates into a set-membership set estimation problem [2] where x(t) is shown to be inside the set X(t) returned by our set-membership observer.
Then we conclude that x(t) cannot be inside the complementary of X(t). This result can be formalized by the following theorem.
Theorem. The virtual intruder has a state vector x(t) inside the set
X(t) = O∩dt·F(X(t−dt)) ∩ \
i
ga−1
i(t)([d(t),∞]), where X(0) = O. As a consequence, the secure zone is
S(t) = projworld(X(t)).
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Figure 1: (left) Paving of the Biscay Bay, (right) Green: Secured zone
Proof (sketch). Two cases should be considered. If no actual intruder exists then S(t) cannot is secured. If the virtual intruder is a real one, its state x(t) is inside X(t) and its position (which is a part of the state) is inside projworld(X(t)). In both situations, the intruder can not be inside S(t).
Method. Each robot follows a reference point. All reference points form a flat ellipsoid which plays the role of a barrier. The strategy is illustrated by Fig 1 for 10 robots. The set O corresponds to the blue area (left). On the right, S(t) is painted green. The observer has been implemented using interval analysis.
References:
[1] L. Jaulin, Mobile robotics, ISTE editions, London, 2016.
[2] M. Kieffer, L. Jaulin, E. Walter, D. Meizel, Nonlinear Identification Based on Unreliable Priors and Data, with Applica- tion to Robot Localization, Robustness in Identification and Con- trol, 3 (1999), LNCIS 245, pp. 190–203.
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